History of Mathematics Project | Babylonian Square Root of 2 Tablet

History of
Mathematics

Project

around 1900–1600 BCE

Babylonian Square Root of 2 Tablet

Computing the square root of 2 in base 60

Circular tablet YPM BC 021354 contains a drawing of a square and its diagonals together with the sexagesimal representation of the diagonal length written in cuneiform numerals.

This famous tablet is one of few Babylonian tablets that consists entirely of a geometrical diagram (plus numbers). It demonstrates that Babylonian scribes both knew the Pythagorean theorem and were able to accurately calculate square roots. The diagram depicts a square with its diagonals together with an accurate approximation of the square root of 2. In addition, the number 30 is written above the upper-left side and the length of a diagonal corresponding to a square of this edge length is written beneath the horizontal diagonal.

Artifact dimensions

7 cm diam.

Original artifact location

Babylonia (historical name), Iraq (current name)

Current artifact location

Peabody Museum of Natural History, New Haven

Catalog number

YPM BC 021354 (current), YBC 07289 (historical)

Timeline

Interactive Content

Computational Explanation

Tablet YPM BC 021354 depicts a square and its diagonals. Based on the cuneiform writing appearing on the tablet, the square is oriented so that the diagonals fall along in compass directions. The upper-left side of the square appears to be labeled, additional digits are written along the horizontal diagonal and a third group of digits appear at a slightly inclined angle below those:

Out[]=

The Babylonians used a base-60 system of numbers in their cuneiform writing script. The following table summarizes cuneiform representations of the whole numbers 1 to 40:

In[]:=

TextGridFlatten/@TransposePartitionTablen,

[◼]	AncientNumberRepresentation

[n,"Babylonian"],{n,40},10,DividersAll

Out[]=

1	11	21	31
2	12	22	32
3	13	23	33
4	14	24	34
5	15	25	35
6	16	26	36
7	17	27	37
8	18	28	38
9	19	29	39
10	20	30	40

As can be seen, the number 10 was represented by a single vertical chevron, the number 20 by two chevrons and the number 30 by three chevrons. The three chevrons in the upper-left portion of the tablet:

Out[]=

… therefore represent the number 30.

To interpret the markings on the middle and lower portions of the tablet, consider a schematic drawing of a square with side lengths a:

Out[]=

By the Pythagorean theorem, the length of a diagonal for a square with side length a is equal to

d



It is straightforward to compute the numerical value of the square root of 2 to any desired number of decimal digits using the Wolfram Language:

In[]:=

,20]

Out[]=

1.4142135623730950488

Based on the position of the numbers along the horizontal diagonal of the tablet:

Out[]=

… it is natural to postulate this sequence of digits might correspond to digits in the Babylonian base-60 expansion of

(i.e. the length of a diagonal of a square with unit side lengths):

In[]:=

DiagonalDigits1=RealDigits[Sqrt[2],60,4][[1]]

Out[]=

{1,24,51,10}

This conjecture is verified by expressing the above sequence of numbers as cuneiform numbers:

[◼]	AncientNumberRepresentation

[DiagonalDigits1,"Babylonian"]

Out[]=





… in agreement with the markings in the central portion of the tablet.

Reconstructing the exact fraction corresponding to the base-60 digits 1, 24, 51, 10 gives:

In[]:=

FromDigits[{DiagonalDigits1,1},60]

Out[]=

30547

21600

… or, more directly:

In[]:=

sqrt2=1+

Out[]=

30547

21600

This fraction has the numeric value:

N[sqrt2,20]

Out[]=

1.4142129629629629630

… which agrees amazingly closely to the numeric expansion of

as previously computed:

In[]:=

,20]

Out[]=

1.4142135623730950488

Since the results of the interpretation of the sequence of digits across of the middle of the table are encouraging, postulate that the inclined sequence of digits:

Out[]=

… corresponds to the diagonal length of a square with edge length 30. Taking three hexadecimal digits of

gives:

In[]:=

DiagonalDigits30=RealDigits[30Sqrt[2],60,3][[1]]

Out[]=

{42,25,35}

… which is written in cuneiform as:

In[]:=

[◼]	AncientNumberRepresentation

[DiagonalDigits30,"Babylonian"]

Out[]=





In this case, the corresponding fractional approximation is:

In[]:=

thirtysqrt2=FromDigits[{DiagonalDigits30,1},60]

Out[]=

30547

720

… which again closely corresponds to the exact value:

In[]:=

N[thirtysqrt2,30

,7]

Out[]=

{42.42639,42.42641}

It is therefore clear that Babylonian scribes not only had knowledge of the special case of the Pythagorean theorem corresponding to an isosceles right triangle, but also that they possessed methods for accurately calculating square roots.

Other Resources

Wolfram Demonstration
Babylonian Numerals

Wolfram Demonstration
Pythagorean Theorem

Additional Reading

Maor, E. The Pythagorean Theorem: A 4000-Year History. Princeton, NJ: Princeton University Press, pp. 4–7, 2007.
Neugebauer, O. and Sachs, A. (Ed.). Mathematical Cuneiform Texts. New Haven, CT: American Oriental Society and the American Schools of Oriental Research, pp. 42–43, 1945.

Additional Links

Image Credits

Yale Peabody Museum of Natural History